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MF-$0.75 HC Not Available from EDRS. PLUS POSTAGE Geometry DOCUMENT RESUME ED 100 648 SE 018 119 AUTHOR Yates, Robert C. TITLE Curves and Their Properties. INSTITUTION National Council of Teachers of Mathematics, Inc., Washington, D.C. PUB DATE 74 NOTE 259p.; Classics in Mathematics Education, Volume 4 AVAILABLE FROM National Council of Teachers of Mathematics, Inc., 1906 Association Drive, Reston, Virginia 22091 ($6.40) EDRS PRICE MF-$0.75 HC Not Available from EDRS. PLUS POSTAGE DESCRIPTORS Analytic Geometry; *College Mathematics; Geometric Concepts; *Geometry; *Graphs; Instruction; Mathematical Enrichment; Mathematics Education;Plane Geometry; *Secondary School Mathematics IDENTIFIERS *Curves ABSTRACT This volume, a reprinting of a classic first published in 1952, presents detailed discussions of 26 curves or families of curves, and 17 analytic systems of curves. For each curve the author provides a historical note, a sketch orsketches, a description of the curve, a a icussion of pertinent facts,and a bibliography. Depending upon the curve, the discussion may cover defining equations, relationships with other curves(identities, derivatives, integrals), series representations, metricalproperties, properties of tangents and normals, applicationsof the curve in physical or statistical sciences, and other relevantinformation. The curves described range from thefamiliar conic sections and trigonometric functions through tit's less well knownDeltoid, Kieroid and Witch of Agnesi. Curve related - systemsdescribed include envelopes, evolutes and pedal curves. A section on curvesketching in the coordinate plane is included. (SD) U S DEPARTMENT OFHEALTH. EDUCATION II WELFARE NATIONAL INSTITUTE OF EDUCATION THIS DOCuME N1 ITASOLE.* REPRO MAE° EXACTLY ASRECEIVED F ROM THE PERSON ORORGANI/AlICIN ORIGIN ATING 11 POINTS OF VIEWOH OPINIONS STATED DO NOT NECESSARILYREPRE INSTITUTE OF SENT OFFICIAL NATIONAL EDUCATION POSITION ORPOLICY $1 loor oiltyi.4410,0 kom niAttintitd.: t .111/11.061 . 141 52.1 "411;2gkE Nn:littitt!riTt" A. , 4 1 44'1,LIAI:iit .11 ,j{j..11:,,stilenhg.,1 L'it!LY II ,.,' itI ,ft ,Mt" )1'14 !!!111%,11,1 ,±14 ROI 911 .14 *f.A Cita. /1" 1!It 1 l' 41 CURVES AND THEIR PROPERTIES CLASSICS IN MATHEMATICS EDUCATION Volume 1: The Pythagorean haposition by Elisha Scott Loomis Volume 2: Number Stories of Long Ago by David Eugene Smith Volume 3: The Trisection Problem by Robert C. Yates Volume 4: Curves and Their Properties by Robert C. Yates RIIRIABLE BESICOP1 CONES AID THE II PROPERTIES OW CAMS THE NATIONAL COUNCIL or TEACHERS OF MATHEMATICS Copyright 1952 by Hobert C. Yates Library of Congress Catalog Card Number: 7.140222 Printed in the United States of America 1974 tV FOREWORD Some mathematical works of considerable vintage havea timeless quality about them. Like classics inany field, they still bring joy and guidance to the reader. Books of this kind, ifno longer readily available, are being sought out by the National Council of Teachers of Mathematics, which has begun to publisha series of such classics. The present title is the fourth volume of the series. A Handbook on Curves and Their Propertieswas first published in 1952 when the author was teaching at the United States Military Academyat West Point. A photoithoprint reproductionwas issued in 1959 by Edwards Brothers, Inc., Lithoprinters, of Ann Arbor,Michigan. The present reprint edition has been similarly produced, by photo-offset, from a copy of the 1959 edition. Except for providing new front matter, includ- ing a biographical sketch of the author and this Foreword byway of, explanation, no attempt has been made to modernize the bookin any way. To do so would surely detract from, rather than add to, its value. ABOUT THE AUTHOR Robert Carl Yates was born in .Falls Church, Virginia,on 10 March 1904. In 1924 he receiveda B.S. degree in civil engineering from Virginia Military Institute. This degreewas followed by an A.B. degree in psy- chology and education from Washington and LeeUniversity in 1926 and by the M.A. and Ph.D. degrees in mathematics andapplied mathematics from Johns Hopkins University in 1928 and 1930. While working on these later degrees Bob Yateswas an instructor at Virginia Military Institute, the University of Maryland,and Johns Hop- kins University. After completing his Ph.D.degree, he accepted a Posi- tion as assistant professor, in 1931, at the University of Maryland,where later he was promoted to associate professor. In 1939 hebecame associate professor of mathematics at Louisiana StateUniversity. As a captain in the Army Reserves, Professor Yates reportedto the United States Military Academy for active dutyon 6 June 1942. Before leaving the Academy herose to the rank of colonel and the title of associate professor of mathematics. He left West Point in August 1954, whena reduction in the number of colonels was authorized at USMA, and accepteda position as professor of mathematics at Virginia PolytechnicInstitute.In 1955 he became professor of mathematics and chairman of the departmentat the College of William and Mary. The last position he heldwas as one of the original professors at the University of South Florida, beginningin 1960. He went to this rew institution as chairman of the Department of Mathematics, resigning the chairmanship in 1962 in order to devotemore time to teach- ing, lecturing, and writing. During his tour of duty at West Point, Dr. Yatesspent many of his summers as a visiting professor. Among the institutions he served were Teachers College, Columbia University; YeshivaUniversity; and johns Hopkins University. Robert Yates wasa man of many talents. Although he was trained in pure and applied mathematics, he became interested In the field of mathematics education rather early in his professionqlcareer. In both vii viii AuouT THE AUTHOR al'eaS he Wilt lip a (Ilse reputation as alecturer and a writer. During his lifetime he had sixty-odd papers published in various researchand mathe- matical-education journals, including the Mathematics Teacher, and in NCTM yearbooks. He also wrote five books dealing with various aspects of geometry, the calculus. and differential equations. From1937 until he was called to active duty atWest Point in 1942, he served on the editorial hoard, and as editor of one department, of the NationalMathematics Magazine. These were some of his professional achievements. Ms activities,how ever, were not limited to theworld of mathematics.At VM1where he was a member of thetrack squad, dramatics and journalism claimed some of his time. Music became a continuing resource. In laterlife his recrea- tions included playing the piano as well as sailing,skating, and golf. Dr. Yates, whose social fraternity was Kappa Alpha, waselected to two scientific honor societies: Gamma Alpha and Sigma Xi. Holding member- ship in the American Mathematical Society, the Mathematical Association of America, and the National Council of Teachersof Mathematics, he was at one period a governor ofthe MAA. He was also a member of the MAA's original ad hoc Committee on the UndergraduateProgram in Mathematics (CUPM ).In late 1961 he was selected by the Association of Higher Education as one of twenty-five"outstanding college andUni- versity educators in America today," and on 4February 1962 he was featured on the ABC-TV program "Meet the Professor." Dr. Yates had been interested in mathematicseducation before 1939. Howcwer, when he came to Louisiana State University, his work inthis field began to expand. Owing to his efforts, the Departmentof Mathe- matics and the College of Education made some importantchanges in the mathematical curriculum for the training of prospective secondaryschool teachers. One of the most important additions was six semesterhours in geometry. Dr. Yates was given this course toteach, and for a text he used his first book, Geometrical Tools, From this beginninghis interest and work in mathematics education increased, while hecontinued to lecture and write in the areas of pure and applied mathematics, The atmosphere at Nk'est Point was quite a change for Dr.Yates. How- ever, even here he continuedhis activities in mathematics education. One of his duties was to supervise and conduct courses inthe techniques of teaching mathematics. These were courses designed forthe groups of new instructors whojoined the departntme staff annually; for most of the faculty at the Academy, then as now, were active-duty officerswho came on a first or second tour ofthree to four years' duration. In performing this duty he was considered a superior instructor and also anexcellent teacher of teachers. ABOUT THE AUTHOR ix After leaving the service Professor Yates continued his effortsto improve mathematics education. During thesummers he taught in several different mathematics institutes, and hewas a guest lecturer in many summer and academic-year institutes supported by the National Science Foundation. In earlier years he both taught and lectured in the grandfather of all institutes, the one developed by Professor W. W. Rankin at Duke Uni- versity.In Virginia and later in Florida he servedas a consultant to teachers of mathematics in various school districts. During the academic years 1961/62 and 1962/63 the University of South Florida was engaged in an experimental television program. Professor Yateswas the television lecturer in the course materials deNeloped through thisprogram. As a result of this programas well as the MAA lectureship program for high schools, supported by he NSF, he traveledto all parts of Florida giving lectures and consulting with high school teachers. Through all these activities Dr. Yates greatly enhancedthe field of mathematics education. He builtup a reputation as an outstanding lec- turer with a pleasing, interest-provoking presentation anda rare ability to talk while illustrating his subject. Those who have heard him will long remember bhp and his great ability. Others will find thathis writ- ings show, soinevAat vicariously, thesesame characteristics. By his first wife, Naomi Sherman, who died in childbirth, he had three children. Robert Jefferson, the eldest, isnow in business in California. Melinda Susan, the youngest, isnow Mrs. Richard B. Shaw, the wife of a Missouri surgeon.
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